Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 497023, 24 pages
doi:10.1155/2012/497023
Research Article
Construction of Optimal Derivative-Free
Techniques without Memory
F. Soleymani,1 D. K. R. Babajee,2 S. Shateyi,3 and S. S. Motsa4
1
Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
Allied Network for Policy Research and Advocacy for Sustainability, IEEE, Mauritius, Mauritius
3
Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
4
School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01,
Pietermaritzburg, South Africa
2
Correspondence should be addressed to S. Shateyi, stanford.shateyi@univen.ac.za
Received 5 July 2012; Accepted 3 September 2012
Academic Editor: Alicia Cordero
Copyright q 2012 F. Soleymani et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Construction of iterative processes without memory, which are both optimal according to the
Kung-Traub hypothesis and derivative-free, is considered in this paper. For this reason, techniques
with four and five function evaluations per iteration, which reach to the optimal orders eight
and sixteen, respectively, are discussed theoretically. These schemes can be viewed as the
generalizations of the recent optimal derivative-free family of Zheng et al. in 2011. This procedure
also provides an n-step family using n 1 function evaluations per full cycle to possess the optimal
order 2n . The analytical proofs of the main contributions are given and numerical examples are
included to confirm the outstanding convergence speed of the presented iterative methods using
only few function evaluations. The second aim of this work will be furnished when a hybrid
algorithm for capturing all the zeros in an interval has been proposed. The novel algorithm could
deal with nonlinear functions having finitely many zeros in an interval.
1. Introduction
The purpose of this study is to present some generalizations of both the celebrated secondorder Steffensen’s method 1, which was advanced by the Danish mathematician Johan
Frederik Steffensen 1873–1961 as follows
−1
xn1 xn − f xn fxn − fxn fxn 2 ,
n 0, 1, 2, . . . ,
1.1
2
Journal of Applied Mathematics
with higher orders of convergence and optimal efficiency indices, and also the extension of
the family of methods in 2. In 1974, Kung and Traub 3 conjectured on the optimality
of multipoint iterative schemes without memory as comes next. An iterative method for
solving single variable nonlinear equation fx 0, with n 1, n ≥ 1, evaluations per
iteration reaches to the maximum order of convergence 2n and the optimal efficiency index
2n/n1 . Consequently, the efficiency of a pth-order method could be given by p1/d , where
d is the whole number of functional evaluations per iteration. Note that Steffensen’s
method possesses 1.414 as its efficiency index. By considering this conjecture, per iteration
the contributed techniques in this research should reach the order 8 with four and order 16
with five evaluations.
The paper unfolds the contents as follows. A collection of pointers to known literature
on derivative-free techniques will be presented in Section 2. This is followed by Section 3,
whereas the central results are given and also by Section 4, where an optimal sixteenth-order
derivative-free technique is constructed. Results and discussion on the comparisons with
other famous derivative-free methods are presented in Section 5, entitled by Computational
Tests. The only difficulty of iterative methods of this type is in the choice of the initial guess.
Using the programming package MATHEMATICA 8, we will provide an algorithm to capture
all the real solutions of a nonlinear equation in a short piece of time by presenting a hybrid
algorithm in Section 6. The conclusions have finally been drawn in Section 7.
2. Selections from the Literature
The literature related to the present paper is substantial, and we do not present a
comprehensive survey. The references of the papers we cite should be consulted for further
reading. Consider iterative techniques for finding a simple root of the nonlinear equation
fx 0, where f : D ⊆ R → R for an open interval D a, b is a scalar function and
it is sufficiently smooth in a neighborhood of α ∈ D. The design of formulas for solving
such equations is an absorbing task in numerical analysis 4. The most frequent approach to
solve the nonlinear equations consists of the implementation of rapidly convergent iterative
methods starting from a reasonably good initial guess to the sought zero of fx.
Many iterative methods have been improved by using various techniques such as
quadrature formulas and weight function; see, for example, 5, 6. All these developments
are aimed at increasing the local order of convergence with a special view of increasing their
efficiency indices. The derivative-involved methods are well discussed in the literature; see,
for example, 7–9 and the references therein. But another thing that should be mentioned is
that for many particular choices of the function f, specially in hard problems, the calculations
of the derivatives are not possible or it takes a deal of time.
Thus, in some situations the considered function fx has an improper behavior or
its derivative is close to 0, which causes that the applied iterative processes fail. That is
why higher-order derivative-free methods are better root solvers and are in focus recently.
The most important merit of Steffensen’s method is that it has quadratic convergence like
Newton’s method. That is, both techniques estimate roots of the equation fx 0 just as
quickly. In this case, quickly means that the number of correct digits in the new obtained
value doubles with each iteration for both. But the formula for Newton’s method requires a
separate function for the derivative; Steffensen’s method does not. Now let us review some
derivative-free techniques.
Journal of Applied Mathematics
3
In 2010, an optimal fourth-order derivative-free method 10 was introduced by Liu et
al. in the following form:
xn1
fxn 2
,
yn xn − f xn fxn − fxn f xn , yn − f yn , zn fxn , zn yn −
f yn ,
fxn , yn 2
2.1
where zn xn fxn . This technique consists of three evaluations of the function per
iteration to obtain the fourth-order convergence. We here remark that fxn , yn , fyn , zn ,
and fxn , zn are divided differences. This scheme has 41/3 ≈ 1.587 as its efficiency index. The
notation of divided differences will be used throughout this paper.
In 2011, Cordero et al. 11 proposed a sixth-order method, which is free from
derivative
2fxn 2
yn xn − ,
f xn fxn − f xn − fxn zn yn −
yn − xn
f yn ,
2f yn − fxn xn1 zn −
2.2
yn − xn
fzn ,
2f yn − fxn and includes 5 evaluations of the function per iteration to reach the efficiency index 61/5 ≈
1.495.
Zheng et al. in 2 presented the following eighth-order derivative-free family without
memory:
yn xn −
fxn ,
fxn , wn wn xn βfxn ,
β ∈ R \ {0},
f yn
,
zn yn − f xn , yn f yn , wn − fxn , wn fzn ,
xn1 zn − f zn , yn f zn , yn , xn zn − yn f zn , yn , xn , wn zn − yn zn − xn which is optimal in the sense of Kung-Traub.
2.3
4
Journal of Applied Mathematics
Soleymani 12 proposed the following optimal three-step iteration without memory,
including four function evaluations, just like 2.3, by using the weight function approach:
fxn ,
wn xn fxn ,
fxn , wn f yn
f yn
f yn
zn yn −
1
,
fxn , wn fxn fwn ⎛
2
f yn
f yn
fzn fzn fzn ⎝
fzn 2
−
zn − 1
fxn f yn
fwn fxn fwn f xn , yn
yn xn −
xn1
3 ⎞
f yn
⎠.
fwn 1
− 30 6fxn , wn 8 fxn , wn 5 fxn , wn 6
2.4
To see a recent paper including derivative-free methods with memory, refer to 13.
For further reading on this topic, refer to 14–19.
3. Construction of a New Eighth-Order Derivative-Free Class
In this paper, we derive a new way for constructing multistep methods of orders eight,
sixteen, and so forth, requiring four, five, and so forth respectively, function evaluations
per iteration. This means that the proposed techniques support the Kung-Traub hypothesis.
Besides, the novel schemes do not use any derivative of a function f whose zeros are sought,
which is another advantage since it is preferable to avoid calculations of derivatives of f in
some situations. Let us consider a two-step cycle in which we have Steffensen’s method in
the first step and Newton’s method in the second step as follows:
fxn 2
,
yn xn − f xn fxn − fxn f yn
xn1 yn − ,
f yn
3.1
with four evaluations, that is, three evaluations of the function fxn , fxn fxn , and fyn and one derivative evaluation f yn per iteration. For simplicity, we assume that fxn fxn fkn ; that is, xn fxn kn . At this time, the main challenge is to approximate
f yn as efficiently as possible such that the fourth-order convergence does not decrease and
the efficiency index increases to 1.587 at the same time. To fulfill this aim, we must use all
of the past three known data, that is, fxn , fkn , and fyn . Now take into account the
following approximation function for ft in the domain D 20:
ft ≈ wt a0 a1 t − xn a2 t − xn 2 a3 t − xn 3 ,
3.2
Journal of Applied Mathematics
5
where its first derivative takes the form w t a1 2a2 t − xn 3a3 t − xn 2 . Clearly, the
unknown three parameters a0 , a1 , and a2 will be obtained by substituting of the known values
in 3.2. Note that a3 is a free real parameter. Hence, we have
a0 fxn ,
fkn , xn − f yn , xn
− a3 xn yn zn ,
a2 kn − yn
a1 f yn , xn − yn − xn a2 a3 xn yn xn zn yn zn .
3.3
Accordingly, by considering a1 , a2 in the derivative form of 3.2, we attain the
following two-step derivative-free technique:
fxn 2
,
kn xn fxn ,
fkn − fxn f yn
yn −
2 , a3 ∈ R,
a1 2a2 yn − xn 3a3 yn − xn
yn xn −
xn1
3.4
wherein there are three evaluations of the function per iteration only. Theorem 3.1 indicates
that the order of 3.4 is four, and hence, it is an optimal derivative-free class with the
efficiency index 1.587. Note that 3.4 is similar to the method given by Ren et al. in 20.
Theorem 3.1. Assume fx to be a sufficiently continuous real function in the domain D. Then the
sequence generated by 3.4 converges to the simple root α ∈ D with fourth-order convergence and it
satisfies the follow-up error equation
en1 c2 c22 − c1 c3 a3 1 c1 2
c13
en4 O en5 ,
3.5
wherein cj f j α/j!, j ≥ 1, and en xn − α.
Proof. See 20.
The given approximation 3.3 of the first derivative of the function in the second step
of our cycle can be applied on any optimal second-order derivative-free method to provide a
new optimal fourth-order method. For example, if one chooses a two-step cycle in which the
first derivative of the function in the first step estimated by backward finite difference, and
after that applies the presented approximation in the second step, then another novel optimal
fourth-order method could be obtained. This will be discussed more in the rest of the work.
6
Journal of Applied Mathematics
Remark 3.2. The simplified form of the approximation for the derivative in the quotient of
Newton’s iteration at the second step of our cycle is as follows:
2
a1 2a2 yn − xn 3a3 yn − xn
fkn , xn − f yn , xn −xn yn
2
f yn , xn 3a3 yn − xn
kn − yn
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn .
3.6
Inspired by the above approach, now we are about to construct new highorder derivative-free methods, which are optimal as well and can be considered as the
generalizations of the methods in 2, 20, 21. To construct such techniques, first we should
consider an optimal fourth-order method in the first and second steps of a multistep cycle.
Toward this end, we take into consideration the following three-step cycle in which Newton’s
method is applied in the third step:
yn xn −
βfxn 2
,
fkn − fxn kn xn βfxn ,
β ∈ R \ {0},
f yn
zn yn − ,
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
xn1 zn −
a3 ∈ R,
3.7
fzn .
f zn It is crystal clear that scheme 3.7 is an eighth-order method with five evaluations per
iteration. The main question is that Is there any way to keep the order on eight but reduce
the number of evaluation while the method be free from derivative? Fortunately, by using all
four past known data, a very powerful approximation of f zn will be obtained. Therefore,
we approximate ft in the domain D, by a new polynomial approximation of degree four
with a free parameter b4 , as follows:
ft ≈ lt b0 b1 t − xn b2 t − xn 2 b3 t − xn 3 b4 t − xn 4 ,
3.8
where its first derivative takes the following form:
f t ≈ l t b1 2b2 t − xn 3b3 t − xn 2 4b4 t − xn 3 .
3.9
Journal of Applied Mathematics
7
Hopefully, we have four known values fxn , fkn , fyn , and fzn . Thus, by
substituting these values in 3.8, we obtain the following linear system of four equations
with four unknowns:
fxn b0 ,
fkn fxn b1 kn − xn b2 kn − xn 2 b3 kn − xn 3 b4 kn − xn 4 ,
2
3
4
f yn fxn b1 yn − xn b2 yn − xn b3 yn − xn b4 yn − xn ,
3.10
fzn fxn b1 zn − xn b2 zn − xn 2 b3 zn − xn 3 b4 zn − xn 4 ,
and consequently we can find the unknown parameters by solving linear system 3.10. We
attain
b0 fxn ,
fkn , xn − fzn , xn /kn − zn − f yn , xn − fzn , xn / yn − zn
b3 kn − yn
− b4 xn yn kn zn
fzn , xn , kn − f zn , xn , yn
− b4 xn yn kn zn ,
kn − yn
b2 f yn , xn , kn − b3 kn − 2xn yn
b4 xn yn xn kn xn zn yn kn yn zn kn zn ,
3.11
b1 fzn , xn − zn − xn 2 b3 − zn − xn b2 .
Now, an efficient, accurate, and optimal eighth-order family, which is free from any
derivative, can be obtained in the following form:
yn xn −
2
β fxn ,
fkn − fxn kn xn βfxn ,
β ∈ R \ {0},
f yn
zn yn − ,
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
xn1 zn −
fzn b1 2b2 zn − xn 3b3 zn − xn 2 4b4 zn − xn 3
,
a3 ∈ R,
b4 ∈ R.
3.12
8
Journal of Applied Mathematics
Remark 3.3. The simplified form of the approximation for the derivative in the quotient of
Newton’s iteration at the third step of our cycle is as comes next:
b1 2b2 zn − xn 3b3 zn − xn 2 4b4 zn − xn 3
fxn , zn f kn , xn , yn − fkn , xn , zn − f yn , xn , zn xn − zn b4 zn − xn zn − kn zn − yn .
3.13
Using Remark 3.3, we can propose the following optimal eighth-order derivative-free
iteration in this paper with some free parameters:
yn xn −
fxn ,
fkn , xn kn xn βfxn ,
β ∈ R \ {0},
f yn
zn yn − ,
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
a3 ∈ R,
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znb4 zn −xnzn −kn zn −yn
3.14
where b4 ∈ R and it consists of four evaluations per iteration to reach the efficiency index
81/4 ≈ 1.682. Theorem 3.4 illustrates its error equation and order of convergence. The main
contribution of this section lies in the following Theorem.
Theorem 3.4. Assume that the function f : D ⊆ R → R has a single root α ∈ D, where D is an
open interval. Then the convergence order of the derivative-free iterative method defined by 3.14 is
eight and it satisfies the following the error equation:
en1 1
c17
4 c22 c22 − c1 c3 a3 1 c1 β c23 − c1 c2 c3 a3 c12 c4 b4 en8 O en9 .
3.15
Proof. To find the asymptotic error constant of 3.14, wherein cj f j α/j!, j ≥ 1, and
en xn − α, we expand any terms of 3.14 around the simple root α in the nth iterate. Thus,
we write fxn c1 en c2 en2 c3 en3 c4 en4 c5 en5 c6 en6 c7 en7 c8 en8 Oen9 . Accordingly, we
attain
yn α c2
1
β en2 · · · O en9 .
c1
3.16
Now we should expand fyn around the simple root by using 3.16. We have
f yn c2 c1 c2 β en2 · · · O en9 .
3.17
Journal of Applied Mathematics
9
Using 3.17 and the second step of 3.14, we attain
f yn
c2
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
1
β en2 · · · O en9 .
c1
3.18
Now, the Taylor expansion of the second step of 3.14, using 3.18, gives us
zn − α 2
c2 c22 − c1 c3 a3 1 c1 β
c13
en4 · · · O en9 .
3.19
A similar Taylor expansion is required for continuing. Hence, it is needed to write the Taylor
expansion of fzn now. Thus, we have
⎛ 2 ⎞
c2 c22 − c1 c3 a3 1 c1 β
⎟ 4
⎜
.
.
.
O
en9 .
fzn ⎝
⎠e
n
c12
3.20
Subsequently for the approximation of f zn , we obtain fxn , zn fkn , xn , yn −
fkn , xn , zn − fyn , xn , zn xn − zn b4 zn − xn zn − kn zn − yn c1 c2 2c23 c12 c4 −
2c1 c2 c3 a3 1 c1 β2 en4 /c13 · · · Oen9 . Furthermore, we attain
fzn fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −zn b4 zn −xn zn −kn zn −yn
2 c2 c22 − c1 c3 a3 1 c1 β en4
·
·
·
O
en9 .
c13
3.21
Using 3.21 and the last step of 3.14, we have the error equation 3.15. This manifests that
3.14 is of optimal order eight with four function evaluations per iteration. Hence, the proof
is complete and 3.14 reaches the efficiency index 81/4 ≈ 1.682.
Noting that 3.14 is an extension of the Steffensen and the Ren et al. methods, we
should here pull the attention toward this fact that 3.14 is also a generalization of the family
given by Zheng et al. in 2. As a matter of fact, our scheme 3.14 in this paper includes two
free parameters more than that of Zheng et al., which shows the generality of our technique.
Clearly, choosing a3 b4 0 will result in the Zheng et al. method 2.4. Thus, it is only an
special element from the family 3.14.
Remark 3.5. The introduced approximation for the first derivative of the function in the third
step of 3.7 can be implemented on any optimal derivative-free fourth-order method without
10
Journal of Applied Mathematics
memory for presenting new optimal eighth-order techniques free from derivative. Namely,
using 2.1 and 3.13 results in the following optimal eighth-order derivative-free method:
fxn ,
kn xn βfxn , β ∈ R \ {0},
fxn , kn f xn , yn − f yn , kn fxn , kn f yn ,
zn yn −
2
f xn , yn
yn xn −
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znγzn −xnzn −kn zn −yn
3.22
where γ ∈ R with the following error equation:
2 1 2
−c
c
c
β
c
β
c
β
1
c
2
c
c
2
1
2
1
3
1
1
2
c17
× −c1 c2 c3 1 c1 β c23 2 c1 β c12 1 c1 β c4 γ en8 O en9 ,
en1
3.23
and if one chooses any of the derivative-free third-order methods without memory in the
first two steps and applied the introduced estimation 3.13, then a sixth-order derivativefree technique will be attained. This also shows that the given approximation doubles the
convergence rate.
To show the generality of the proposed class and by using Remark 3.5, in what follows,
we give some other optimal three-step four-point root solvers without memory. For the first
example and by using 11 in 13, we have
fxn ,
kn xn βfxn , β ∈ R \ {0},
fxn , kn f yn
f yn
f yn
zn yn −
1
,
fxn , kn fxn fkn yn xn −
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znγzn −xnzn −kn zn −yn
3.24
Journal of Applied Mathematics
11
where γ ∈ R and
en1 xn1 − α 1
c17
c2 c1 c2 β
−c1 c3 1 c1 β c22 5 c1 β 5 c1 β
2 8
en O en9 .
× −c1 c2 c3 1 c1 β c23 5 c1 β 5 c1 β c12 1 c1 β c4 γ
3.25
Using a different fourth-order method according to 13, we get that
fxn ,
kn xn βfxn , β ∈ R \ {0},
fxn , kn f yn
1
zn yn −
,
fxn , kn 1 − f yn /fxn − f yn /fkn yn xn −
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znγzn −xnzn −kn zn −yn
3.26
wherein γ ∈ R, with the following error relation:
en1 4 c22 c22 − c1 c3 1 c1 β c23 − c1 c2 c3 c12 c4 γ
c17
en8 O en9 .
3.27
We can also propose the following general family of three-step iterations using 22:
fxn ,
kn xn βfxn , β ∈ R \ {0},
fxn , kn f yn
2 βfxn , kn f yn
zn yn −
1
,
fxn , kn 1 βfxn , kn fxn yn xn −
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znγzn −xnzn −kn zn −yn
3.28
12
Journal of Applied Mathematics
where γ ∈ R
2 1 −c1 c3 1 c1 β c22 5 c1 β 5 c1 β
c2 c1 c2 β
7
c1
× −c1 c2 c3 1 c1 β c23 5 c1 β 5 c1 β c12 1 c1 β c4 γ en8 O en9 .
en1
3.29
Remark 3.6. Using backward finite difference approximation for the first derivative of the
function in the first step of our cycles will end in other new methods as comes next:
fxn ,
kn xn − βfxn , β ∈ R \ {0},
fxn , kn f xn , yn − f yn , kn fxn , kn f yn ,
zn yn −
2
f xn , yn
yn xn −
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znγzn −xnzn −kn zn −yn
3.30
where γ ∈ R
2 1 2
2
c
c
β
c
β
c
β
−1
c
1
−
c
−2
c
1
1
3
1
1
2
2
c17
8
× c1 c2 c3 1 − c1 β c23 −2 c1 β c12 −1 c1 β c4 γ
en O en9 .
en1
3.31
We can also have
yn xn −
fxn ,
fxn , kn kn xn − βfxn ,
f yn
β ∈ R \ {0},
zn yn − ,
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
a3 ∈ R,
xn1
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znγzn −xnzn −kn zn −yn
3.32
Journal of Applied Mathematics
13
with the following error equation γ ∈ R:
4
1 2 2
c
c
−
c
a
−1 c1 β
c
1
3
3
2
2
7
c1
8
en O en9 .
× c23 − c1 c2 c3 a3 c12 c4 γ
en1
3.33
4. Higher-Order Optimal Schemes
In this section, we take a special heed to generalize the novel scheme 3.14 by using the same
idea. To increase the local order of convergence and the efficiency index more, we should
consider cycles in which there are four, five, and so forth steps. Here, we consider a four-step
cycle at which 3.14 is in the first three steps and Newton’s method is in the fourth step as
follows a3 , b4 ∈ R:
yn xn −
fxn ,
fkn , xn kn xn βfxn ,
β ∈ R \ {0},
f yn
zn yn − ,
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
wn
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znb4 zn −xnzn −kn zn −yn
xn1 wn −
fwn .
f wn 4.1
It is easy to check that 4.1 is a sixteenth-order method with six evaluations per
iteration and it has the efficiency index 161/6 ≈ 1.587. To make 4.1 optimal derivative-free
multipoint technique with 161/5 ≈ 1.741 as the efficiency index, it is required to approximate
f wn effectively. To do this, we estimate ft in the domain D, by a novel polynomial of
degree four like the similar cases in 3.2 and 3.8 as follows:
ft ≈ rt r0 r1 t − xn r2 t − xn 2 r3 t − xn 3 r4 t − xn 4 r5 t − xn 5 ,
4.2
14
Journal of Applied Mathematics
where its first derivative takes the form as follows: f t ≈ r t r1 2r2 t − xn 3r3 t − xn 2 4r4 t − xn 3 5r5 t − xn 4 . Substituting the known values in 4.2 gives us the unknown values
by solving a system of linear equations in what follows:
⎞⎛ ⎞
kn − xn kn − xn 2 kn − xn 3 kn − xn 4
r
2 3 4 ⎟ 1
⎜
⎜
yn − xn
yn − xn
yn − xn ⎟⎜r2 ⎟
⎜ yn − xn
⎟
⎟
⎜
2
3
⎝ zn − xn zn − xn zn − xn zn − xn 4 ⎠⎝r3 ⎠
r4
wn − xn wn − xn 2 wn − xn 3 wn − xn 4
⎞
⎛
fkn − fxn − r5 kn − xn 4
4 ⎟
⎜
⎜ f yn − fxn − r5 yn − xn ⎟
⎜
⎟.
4
⎝ fzn − fxn − r5 zn − xn ⎠
fwn − fxn − r5 wn − xn 4
⎛
4.3
Note that for the case of multipoint methods, it is more desirable to apply the symbolic
calculations using one of the modern software in mathematics. Using Mathematica 8 gives us
the unknown parameters by the command LinearSolve as in Algorithm 1.
Here to cut the long story short, we just provide the following theorem.
Theorem 4.1. The four-step method
yn xn −
fxn ,
fkn , xn kn xn βfxn ,
β ∈ R \ {0},
f yn
zn yn − ,
f yn , xn f kn , xn , yn yn − xn a3 yn − xn yn − kn
wn
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znb4 zn −xnzn −kn zn −yn
xn1 wn −
fwn r1 2r2 wn − xn 3r3 wn − xn 2 4r4 wn − xn 3 5r5 wn − xn 4
,
r5 ∈ R,
4.4
converges to the simple root of fx 0 in the domain D with local sixteenth order of convergence,
where the first three steps are the optimal eighth-order method 3.14 and r1 , r2 , r3 and r4 are given as
follows:
fzn , xn − fwn , xn fkn , xn − fzn , xn kn − wn kn − yn kn − zn kn − wn wn − yn wn − zn f yn , xn − fzn , xn ,
kn − yn wn − yn yn − zn
fkn , xn − fzn , xn /kn − zn −f yn , xn fzn , xn / yn − zn
,
r3 kn − yn
r4 Journal of Applied Mathematics
15
2
3
−fkn , xn f yn , xn kn − xn 2 r3 kn − xn 3 r4 − r3 xn − yn r4 xn − yn
r2 ,
yn − kn
r1 fwn , xn − r2 wn − xn − r3 wn − xn 2 − r4 wn − xn 3 .
4.5
Proof. The proof of this theorem is similar to the proofs of Theorems 3.1 and 3.4. Hence, it is
omitted.
Remark 4.2. The simplified form of the approximation for the derivative in the quotient of
Newton’s iteration at the fourth step of our cycle without the index n is as comes next:
r1 2r2 wn − xn 3r3 wn − xn 2 4r4 wn − xn 3 5r5 wn − xn 4
fzn , wn f wn , zn , yn wn − zn f wn , zn , yn , xn wn − zn wn − yn
f wn , zn , yn , xn , kn × wn − zn wn − yn wn − xn r5 wn − xn wn − kn wn − yn wn − zn Υn.
4.6
Remark 4.3. Note that if we choose another optimal derivative-free eighth-order method in
the first three steps of 4.4 and then implement the introduced approximation in the fourthstep, we will obtain a new technique of optimal sixteenth convergence rate to solve nonlinear
scalar equations. In what follows, we give some of such optimal 16th -order schemes:
fxn ,
kn xn βfxn , β ∈ R \ {0},
fkn , xn f xn , yn − f yn , kn fxn , kn f yn ,
zn yn −
2
f xn , yn
yn xn −
wn
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znb4zn −xnzn −kn zn −yn
xn1 wn −
fwn ,
Υn
where b4 , r5 ∈ R and
fxn ,
kn xn − βfxn , β ∈ R \ {0},
fkn , xn f xn , yn − f yn , kn fxn , kn zn yn −
f yn ,
2
f xn , yn
yn xn −
4.7
16
Journal of Applied Mathematics
wn
zn −
fzn ,
fxn , zn f kn , xn , yn −fkn , xn , zn −f yn , xn , zn xn −znb4 zn −xnzn −kn zn −yn
xn1 wn −
fwn .
Υn
4.8
Remark 4.4. If one considers an n-step derivative-free construction in which the first n−1 steps
are any of the existing optimal derivative-free methods of order 2n−1 , then by considering a
polynomial approximation of degree n 1 with one free parameter, as in the idea above, as
discussed above, an optimal derivative-free method of order 2n with the optimal efficiency
index 2 will be attained.
Now, let us list the efficiency indices of some of the well-known methods with the
contributed schemes in this study. The results of comparisons are made in Table 1. As it
shows, our schemes are totally better than the existing methods in the literature in terms
of efficiency index.
5. Computational Tests
The practical utilities of the main contributions in this paper are given by solving a couple
of numerical examples and comparison with other well-known methods of different orders
in this section. The main purpose of demonstrating the new derivative-free methods for
nonlinear equations is purely to illustrate the accuracy of the approximate solution, the
stability of the convergence, and the consistency of the results and to determine the efficiency
of the new iterative techniques.
We have computed the root of each test function for an initial approximation in the
neighborhood D. We remark that we have chosen for comparison only the methods which
do not require the computation of derivatives of the function to carry out iterations. We have
used the fourth-order scheme of Ren et al. RM4, the eighth-order derivative-free family of
Zheng et al. 2.3 with β 1, the eighth-order derivative-free method of Soleymani 2.4, and
our novel contributed techniques 3.14 with a3 1, b4 −3, 3.22 with γ 0, and 3.30
with γ 0. We here recall the family of derivative-free methods, which was given by Kung
and Traub KT8 3 as comes next:
yn xn βfxn ,
fxn f yn
,
zn yn − β f yn − fxn fxn f yn
1
1
wn zn −
− ,
fzn − fxn f yn , xn
f zn , yn
Journal of Applied Mathematics
17
fxn f yn fzn xn1 wn −
fwn − fxn 1
1
1
1
1
1
− ×
−
− ,
fzn −fxn f zn , yn f yn , xn
fwn −f yn fwn , zn f zn , yn
5.1
where β ∈ R \ {0}; we have used this scheme in numerical comparison by choosing β 1. It
is completely clear that 4.4 performs better than any of the other methods; hence we do not
include it in numerical comparisons. The test functions and their roots are given in Table 2.
The results are summarized in Tables 3 and 4, in terms of the required number of
iterations to obtain an approximation of the root. In our numerical comparisons, we have
used 2000-digit floating point in our calculations.
As Tables 3 and 4 manifest, our contributed methods are accurate and efficient in
contrast by the other high-order schemes in terms of required number of iterations. It
is well known that convergence of iteration formula is guaranteed only when the initial
approximation is sufficiently near to the root. This issue will be the main body of the
next section. However, our class of derivative-free method is much better and has better
convergence rate in contrast to the existing high-order derivative-free methods in the
literature.
6. A Hybrid Algorithm to Capture All the Solutions
An important aspect in implementing the iterative methods for the solution of nonlinear
equations and systems relies on the choice of the initial approximation. There are a few
known ways in the literature 23 to extract a starting point for the solutions of nonlinear
functions. In practice, users need to find out robust approximations for all the zeros in an
interval. Thus, to remedy this and to respond on this need, we provide a way to extract all
the real zeros of nonlinear function in the interval D a, b. We use the command Reduce
in MATHEMATICA 8 24, 25. Hence, we give a hybrid algorithm including two main steps,
a predictor and a corrector. In the predictor step, we extract initial approximations for all
the zeros in an interval up to 8 decimal places. Then in the corrector step, an eighth-order
method, for example, 3.28, will be used to boost up the accuracy of the starting points up to
any tolerance. We also give some significant cautions for applying on different test functions.
In what follows, we keep going by choosing an oscillatory function fx − cos2 −
x2 logx/7 1/10 in the domain D 0, 15.
Let us define the function and the domain for imposing the Reduce command as in
Algorithm 2.
One may note that Reduce works with function of exact arithmetic. Hence, if a
nonlinear function is the floating point arithmetic, that is, has inexact coefficients, thus we
should write it in the exact arithmetic when we enter it into the above piece of code.
Now we store the list of initial approximations in initialValues, by the following
piece of code, which also sort the initial points. The tol will specify that the accuracy
of each member of the provided sequence to be correct up to utmost tol, decimal places
Algorithm 3.
18
Journal of Applied Mathematics
LinearSolve{{k-x,((k-x)∧ 2), ((k-x)∧ 3),((k-x)∧ 4) } ,
{ y-x, ((y-x)∧ 2),((y-x)∧ 3), ((y-x)∧ 4) } , { z-x, ((z-x)∧ 2),
((z-x)∧ 3),((z-x)∧ 4) } , { w-x, ((w-x)∧ 2), ((w-x)∧ 3),((w-x)∧ 4)}},
{ fk-fx-r5∗(k-x)∧ 4,fy]-fx-r5∗(y-x)∧ 4, fz-fx
-r5∗(z-x)∧ 4,fw-f[x-r5∗(w-x)∧ 4}//FullSimplify
Algorithm 1
ClearAllf, x, rts, initialValues, tol, a, b, j,
NumberOfGuesses, FD, nMax, beta, gamma, y, z, k,
fx, fk, fy, fz
fx := Logx/7 - Cosx∧ 2 - 2 + 1/10
a = 0; b = 15;
Plotfx, { x, a, b } , Background -> LightBlue,
PlotStyle -> {Brown, Thick}, PlotRange -> All,
PerformanceGoal -> "Quality"
rts = Reducefx == 0 && a < x < b, x;
Algorithm 2
It is obvious that f is so oscillatory, and by the above predictor piece of Mathematica
code, we attain that it has 69 real solutions. Note that the graph of the function f has been
drawn in Figure 1. Now it is time to use one of the derivative-free methods given in Section 2
or 3 for correcting the elements of the obtained list. We use the method 3.28 in Algorithm 4.
In Algorithm 4, we have the ability to choose the nonzero free parameter β, as well as
the parameter gamma. Note that for optimal eighth-order methods and with such an accurate
list of initial points, we believe that only two full iterations are required. Hence, we have
selected nMax 2. Note that if a user needs much more accuracy, thus higher number of steps
should be taken. It should be remarked that in order to work with such a high accuracy, we
must then choose more than 2000 decimal places arithmetic in our calculations.
Due to page limitations, we are not able to list the results for all 69 zeros. However,
running the above algorithm could capture all the real zeros of the nonlinear functions.
We also remark three points. One is that for many oscillatory function or for nonsmooth
functions, the best way is to first divide the whole interval into some subintervals and then
find all the zeros of the function on the subintervals. And second, in case of having a root
cluster, that is, when the zeros are concentrated on a very small area, then it would be better to
increase the first tolerance of our algorithm in the predictor step, to find reliable starting points
and then start the process. And last, if the nonlinear function has an exact solution, that is to
say, an integer be the solution of a nonlinear function, then the first step of our algorithm finds
this exact solution, and an error-like message would be generated by applying our second
step.
For instance, the function gx x2 − 4 sin100x on the interval D 0, 10 has 319
real solutions in which one of them its plot is given in Figure 2, that is, 2, is an exact one.
Journal of Applied Mathematics
19
tol = 8;
initialValues = SortNx /. { ToRulesrts}, tol
LengthinitialValues
AccuracyinitialValues
Algorithm 3
Table 1: Comparisons of efficiency indices for different types of derivative-free schemes.
1.1
2.1
2.2
2.3
2.4
5.1
3.14
4.4
M.
1/2
1/3
1/5
1/4
1/4
1/4
1/4
1/5
2
≈
1.414
4
≈
1.587
6
≈
1.495
8
≈
1.682
8
≈
1.682
8
≈
1.682
8
≈
1.682
16
≈ 1.741
E.I.
Table 2: The test functions considered in this study.
Test functions
Simple zeros
Initial guesses
f1 x 3x sinx − ex
α1 ≈ 0.360421702960324
x0 0.2
f2 x sinx − 0.5
α2 ≈ 0.523598775598299
x0 0.3
f3 x x − e − 3x 2
α3 ≈ 0.257530285439861
x0 0.4
f4 x x3 4x2 − 10
α4 ≈ 1.365230013414097
x0 1.37
f5 x xe−x − 0.1
α5 ≈ 0.111832559158963
x0 0.2
f6 x x − 10
α6 ≈ 2.15443490031884
x0 2.16
f7 x 10xe−x − 1
α7 ≈ 1.679630610428450
x0 1.4
f8 x cosx − x
α8 ≈ 0.739085133215161
x0 0.3
2
x
3
2
Table 3: Results of comparisons for different methods after two full iterations.
Test functions
f1 x2 f2 x2 f3 x2 f4 x2 f x 5 2 f6 x2 f x 7 2 f8 x2 RM4
0.4e − 13
0.1e − 14
0.4e − 20
0.1e − 28
0.6e − 15
0.7e − 29
0.1e − 4
0.1e − 15
2.3
0.1e − 57
0.4e − 64
0.1e − 83
0.1e − 124
0.1e − 59
0.4e − 125
0.2e − 24
0.2e − 71
2.4
0.4e − 45
0.7e − 51
0.2e − 82
0.1e − 94
0.1e − 34
0.1e − 95
0.1e − 14
0.1e − 62
5.1
0.2e − 52
0.2e − 57
0.7e − 82
0.4e − 115
0.6e − 49
0.2e − 115
0.5e − 9
0.1e − 58
3.14
0.1e − 44
0.7e − 41
0.2e − 61
0.1e − 118
0.1e − 53
0.2e − 117
0.3e − 26
0.2e − 44
3.22
0.2e − 57
0.1e − 63
0.5e − 83
0.6e − 124
0.1e − 53
0.2e − 124
0.1e − 33
0.3e − 78
3.30
0.2e − 65
0.1e − 96
0.5e − 79
0.2e − 126
0.1e − 73
0.1e − 127
0.1e − 16
0.1e − 34
Table 4: Results of comparisons for different methods after three full iterations.
Test functions
RM4
f1 x3 0.3e − 54
f2 x3 0.1e − 59
f3 x3 0.2e − 84
f4 x3 0.2e − 117
|f5 x3 |
0.6e − 60
f6 x3 0.1e − 118
f x 0.9e − 20
7 3 f8 x3 0.3e − 66
2.3
0.1e − 466
0.5e − 516
0.1e − 676
0.9e − 1004
0.5e − 478
0.1e − 1008
0.4e − 199
0.1e − 578
2.4
0.1e − 363
0.1e − 408
0.3e − 668
0.2e − 761
0.2e − 275
0.3e − 768
0.3e − 119
0.3e − 507
5.1
0.1e − 422
0.9e − 462
0.1e − 663
0.3e − 927
0.7e − 391
0.7e − 930
0.1e − 75
0.1e − 476
3.14
0.3e − 361
0.7e − 328
0.1e − 497
0.1e − 955
0.1e − 427
0.2e − 946
0.2e − 214
0.3e − 360
3.22
0.5e − 463
0.1e − 510
0.1e − 673
0.2e − 999
0.2e − 429
0.1e − 1002
0.2e − 273
0.1e − 634
3.30
0.6e − 529
0.4e − 780
0.2e − 640
0.6e − 1019
0.7e − 591
0.4e − 1030
0.1e − 136
0.4e − 283
20
Journal of Applied Mathematics
nMax = 2; beta = SetAccuracy1, 2000;
gamma = SetAccuracy0, 2000;
Forj = 1, j <= NumberOfGuesses, j++,
{x = SetAccuracyinitialValuesj, 2000,
Do
fx = SetAccuracyfx, 2000;
k = SetAccuracyx + beta fx, 2000;
fk = SetAccuracyfk, 2000;
FD = SetAccuracy(fk - fx)/(beta fx), 2000;
y = SetAccuracyx - fx/FD, 2000;
fy = SetAccuracyfy, 2000;
z = SetAccuracyy - (fy/FD)∗(1 + ((2 + beta FD)/(1
+ beta FD)) (fy/fx)), 2000;
fz = SetAccuracyfz, 2000;
FDxz = SetAccuracy(fz - fx)/(z - x), 2000;
FDxy = SetAccuracy(fx - fy)/(x - y), 2000;
FDxk = SetAccuracy(fk - fx)/(k - x), 2000;
x = SetAccuracyz - fz/(FDxz + ((FDxk - FDxy)/(k - y)
- (FDxk - FDxz)/(k - z)
- (FDxy - FDxz)/(y - z))∗(x - z)
+ gamma (z - x)∗(z - k)∗(z - y)), 2000;
, {n, nMax};
PrintColumn{
Nx, 128,
Nfx}
, Background -> {{LightGreen, LightGray}},
Frame -> True]];
}
];
Algorithm 4
Thus, the first step of the previously mentioned algorithm finds the following very efficient
list of starting points in which 2, is the exact solution:
{0.031415927, 0.062831853, 0.094247780, 0.12566371, 0.15707963, 0.18849556, 0.21991149,
0.25132741, 0.28274334, 0.31415927, 0.34557519, 0.37699112, 0.40840704, 0.43982297,
0.47123890, 0.50265482, 0.53407075, 0.56548668, 0.59690260, 0.62831853, 0.65973446,
0.69115038, 0.72256631, 0.75398224, 0.78539816, 0.81681409, 0.84823002, 0.87964594,
0.91106187, 0.94247780, 0.97389372, 1.0053096, 1.0367256, 1.0681415, 1.0995574,
1.1309734, 1.1623893, 1.1938052, 1.2252211, 1.2566371, 1.2880530, 1.3194689, 1.3508848,
1.3823008, 1.4137167, 1.4451326, 1.4765485, 1.5079645, 1.5393804, 1.5707963, 1.6022123,
1.6336282, 1.6650441, 1.6964600, 1.7278760, 1.7592919, 1.7907078, 1.8221237, 1.8535397,
1.8849556, 1.9163715, 1.9477874, 1.9792034, 2.0000000, 2.0106193, 2.0420352, 2.0734512,
2.1048671, 2.1362830, 2.1676989, 2.1991149, 2.2305308, 2.2619467, 2.2933626, 2.3247786,
Journal of Applied Mathematics
21
2.3561945, 2.3876104, 2.4190263, 2.4504423, 2.4818582, 2.5132741, 2.5446900, 2.5761060,
2.6075219, 2.6389378, 2.6703538, 2.7017697, 2.7331856, 2.7646015, 2.7960175, 2.8274334,
2.8588493, 2.8902652, 2.9216812, 2.9530971, 2.9845130, 3.0159289, 3.0473449, 3.0787608,
3.1101767, 3.1415927, 3.1730086, 3.2044245, 3.2358404, 3.2672564, 3.2986723, 3.3300882,
3.3615041, 3.3929201, 3.4243360, 3.4557519, 3.4871678, 3.5185838, 3.5499997, 3.5814156,
3.6128316, 3.6442475, 3.6756634, 3.7070793, 3.7384953, 3.7699112, 3.8013271, 3.8327430,
3.8641590, 3.8955749, 3.9269908, 3.9584067, 3.9898227, 4.0212386, 4.0526545, 4.0840704,
4.1154864, 4.1469023, 4.1783182, 4.2097342, 4.2411501, 4.2725660, 4.3039819, 4.3353979,
4.3668138, 4.3982297, 4.4296456, 4.4610616, 4.4924775, 4.5238934, 4.5553093, 4.5867253,
4.6181412, 4.6495571, 4.6809731, 4.7123890, 4.7438049, 4.7752208, 4.8066368, 4.8380527,
4.8694686, 4.9008845, 4.9323005, 4.9637164, 4.9951323, 5.0265482, 5.0579642, 5.0893801,
5.1207960, 5.1522120, 5.1836279, 5.2150438, 5.2464597, 5.2778757, 5.3092916, 5.3407075,
5.3721234, 5.4035394, 5.4349553, 5.4663712, 5.4977871, 5.5292031, 5.5606190, 5.5920349,
5.6234508, 5.6548668, 5.6862827, 5.7176986, 5.7491146, 5.7805305, 5.8119464, 5.8433623,
5.8747783, 5.9061942, 5.9376101, 5.9690260, 6.0004420, 6.0318579, 6.0632738, 6.0946897,
6.1261057, 6.1575216, 6.1889375, 6.2203535, 6.2517694, 6.2831853, 6.3146012, 6.3460172,
6.3774331, 6.4088490, 6.4402649, 6.4716809, 6.5030968, 6.5345127, 6.5659286, 6.5973446,
6.6287605, 6.6601764, 6.6915924, 6.7230083, 6.7544242, 6.7858401, 6.8172561, 6.8486720,
6.8800879, 6.9115038, 6.9429198, 6.9743357, 7.0057516, 7.0371675, 7.0685835, 7.0999994,
7.1314153, 7.1628313, 7.1942472, 7.2256631, 7.2570790, 7.2884950, 7.3199109, 7.3513268,
7.3827427, 7.4141587, 7.4455746, 7.4769905, 7.5084064, 7.5398224, 7.5712383, 7.6026542,
7.6340701, 7.6654861, 7.6969020, 7.7283179, 7.7597339, 7.7911498, 7.8225657, 7.8539816,
7.8853976, 7.9168135, 7.9482294, 7.9796453, 8.0110613, 8.0424772, 8.0738931, 8.1053090,
8.1367250, 8.1681409, 8.1995568, 8.2309728, 8.2623887, 8.2938046, 8.3252205, 8.3566365,
8.3880524, 8.4194683, 8.4508842, 8.4823002, 8.5137161, 8.5451320, 8.5765479, 8.6079639,
8.6393798, 8.6707957, 8.7022117, 8.7336276, 8.7650435, 8.7964594, 8.8278754, 8.8592913,
8.8907072, 8.9221231, 8.9535391, 8.9849550, 9.0163709, 9.0477868, 9.0792028, 9.1106187,
9.1420346, 9.1734505, 9.2048665, 9.2362824, 9.2676983, 9.2991143, 9.3305302, 9.3619461,
9.3933620, 9.4247780, 9.4561939, 9.4876098, 9.5190257, 9.5504417, 9.5818576, 9.6132735,
9.6446894, 9.6761054, 9.7075213, 9.7389372, 9.7703532, 9.8017691, 9.8331850, 9.8646009,
9.8960169, 9.9274328, 9.9588487, 9.9902646}.
6.1
22
Journal of Applied Mathematics
2
4
6
8
10
12
14
−5
−10
−15
Figure 1: The plot of the function fx including the zeros.
100
50
2
4
6
8
10
−50
−100
Figure 2: The plot of the function gx including the zeros.
Now we are able to solve nonlinear equations with finitely many roots in an interval
and find all the real zeros in a short piece of time. Finding robust ways, to capture the complex
solutions along working with complex nonlinear functions, can be taken into account as
future works.
7. Concluding Comments
In a word, a general way for producing optimal methods based upon the still unproved
hypothesis of Kung-Traub 3 has been discussed fully. In these methods, it is necessary to
begin with one initial approximation x0 . The convergence results of the iterative sequences
generated by new algorithms were provided. Our contributed techniques without memory
reach the optimal efficiency indices 81/4 ≈ 1.682 and 161/5 ≈ 1.741.
In the limitation case, based on a polynomial of degree n 1 for a method with n
steps, we can obtain an iterative scheme with efficiency index 2, which is the highest possible
efficiency in this field of study for methods without memory. By numerical experiments
above, we verify that new methods are effective and accurate. We conclude that the new
methods presented in this paper are competitive with other recognized efficient equation
solvers.
Note that construction of some ways, which launch n-step Steffensen-type methods
with memory with the order up to 2n 2n−1 50% of an improvement requiring the same
Journal of Applied Mathematics
23
computational cost and applying elegant and quite natural accelerating procedure, could be
considered as future works. Note that the second aim of this paper was given by presenting
a robust algorithm for solving nonlinear equations with finitely many roots in an interval. In
fact, a way for extracting very robust initial approximations as the seeds to start the iterative
Steffensen-type methods was presented. Now we are able to capture all the solutions with
any desire number of decimal places.
Acknowledgment
The second author is an IEEE member.
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